Can romex wire run from wall plug to light switch and then to light attaced by staples to interior face of logs of a log home.

NO, Dangerous and illegal.
Romex is not approved for use exposed like that. Use Wiremold products type surface raceway

Wiremold Cable Raceway | Steel, Wall & Floor Cable Management | by Legrand

Well, romex is explicitly allowed to be run exposed, but it cannot be run where exposed to physical damage.
Exposed down low on the interior of a log home would definitely be considered exposed to damage.